In the spherical-harmonics component form of the Boltzmann equation for a cylindrical system with radial symmetry and z-dependence, application of the commonly applied boundary conditions gives rise to more equations than there are unknowns. This paper considers selected combinations of the equations resulting from the application of these radial boundary conditions for the P3 approximation. Numerical evaluation of the angular flux at a free surface and at an interface between two different media shows that a combination of equations which closely satisfies the desired conditions is that suggested by Davison.